A Correctly Rounded Mixed-Radix Fused-Multiply-Add
Bibliographic record
Abstract
The IEEE 754-2008 Standard governs Floating-Point Arithmetic in all types of Computer Systems. The Standard provides for two radices, 2 and 10. It specifies conversion operations between these radices, but does not allow floating-point formats of different radices to be mixed in computational operations. In contrast, the Standard does provide for mixing formats of one radix in one operation. In order to enhance the Standard and make it closed under all basic computational operations, we propose an algorithm for a correctly rounded mixed-radix Fused-Multiply-and-Add (FMA). Our algorithm takes any combination of IEEE754 binary64 and decimal64 numbers in argument and provides a result in IEEE754 binary64 and decimal64, rounded according to any for the five IEEE754 rounding modes. Our implementation does not require any dynamic memory allocation; its runtime can be bounded statically. We compare our implementation to a basic mixed-radix FMA implementation based on the GMP Multiple Precision library.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.018 | 0.010 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".